Let K be a field of positive characteristic p and KG the group algebra of a group G. It is known that, if KG is Lie nilpotent, then its upper (and lower) Lie nilpotency index is at most |G | + 1, where |G | is the order of the commutator subgroup. The authors previously determined those groups G for which this index is maximal and here they determine the groups G for which it is 'almost maximal', that is, it takes the next highest possible value, namely |G | − p + 2.
We characterize the restricted Lie algebras L whose restricted universal enveloping algebra u(L) is Lie metabelian. Moreover, we show that the last condition is equivalent to u(L) being strongly Lie metabelian
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